Rejoinder on: Minimal penalties and the slope heuristics: a survey

This text is the rejoinder following the discussion of a survey paper about minimal penalties and the slope heuristics (Arlot, 2019. Minimal penalties and the slope heuristics: a survey. Journal de la SFDS). While commenting on the remarks made by the discussants, it provides two new results about the slope heuristics for model selection among a collection of projection estimators in least-squares fixed-design regression. First, we prove that the slope heuristics works even when all models are significantly biased. Second, when the noise is Gaussian with a general dependence structure, we compute expectations of key quantities, showing that the slope heuristics certainly is valid in this setting also.